Consider the intial value problem y' -\frac{3}{2}y=e^{t}, y(0)=y_{0} .
(a) find the value of y_{0} that seperates solutions that grow positively as t \rightarrow infinity from those that groe negativity.
(b) how does the solution that corresponds to the critical value of y_{0} behave as t \rightarrow infinity .

37

step-by-step explanation:

it is x=-4

step-by-step explanation:

x =-.20752

step-by-step explanation:

4^(x+2)=12

4^(x+2)=12

take the log base 4 on each side

log 4 (4^(x+2) = log 4(12)

we know log b(a^y) = y log b (a)

(x+2) log 4(4) = log 4 (12)

x+2 = log 4 (12)

we know   logb c = loga c/loga b

we want to convert to base 10   so a = 10

by default, we do not write the 10   c = 4   and b = 10

log4 10 = log 12/log 4

log4 10 = log 12/ log 4

x+2 = log 12/ log 4

subtract 2 from each side

x +2 -2 = log 12/ log 4 -2

x = 1.79248125 -2

x =-.20752